document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=515234>Question 515234</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Nite Time Inn has a toll free telephone number so that customers can call at any time to make a reservation.  A typical call takes about 4 minutes to complete, and the time required follows an exponential distribution. Find the probability that :<BR>\n" );
document.write( "(a) a call takes 3 minutes or less <BR>\n" );
document.write( "(b) a call takes 4 minutes or less <BR>\n" );
document.write( "(c) a call takes 5 minutes or less <BR>\n" );
document.write( "(d) a call takes longer than 5 minutes </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #343866 by </B><A HREF=/tutors/aboutme.mpl?userid=drcole><B>drcole(72)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drcole><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=343866\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> An exponential distribution with parameter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lambda ALIGN=MIDDLE  ALT=\"lambda\"   DISABLED_style= \"max-width:90%; height:auto;\" > tell us the probability that some process will last a given amount of time or less, given that we know that the mean length of such a process is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flambda ALIGN=MIDDLE  ALT=\"1%2Flambda\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  In this case, we know that a typical call takes about 4 minutes to complete, so\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1%2Flambda+=+4+ ALIGN=MIDDLE  ALT=\"+1%2Flambda+=+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lambda+=+1%2F4+=+0.25+ ALIGN=MIDDLE  ALT=\"+lambda+=+1%2F4+=+0.25+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (taking the reciprocal of both sides)\r<BR>\n" );
document.write( "\n" );
document.write( "Given <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lambda ALIGN=MIDDLE  ALT=\"lambda\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the probability that a process will last T minutes or less is given by the cumulative distribution function:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28time+%3C=+T%29+=+1+-+e%5E%28-lambda%2AT%29 ALIGN=MIDDLE  ALT=\"P%28time+%3C=+T%29+=+1+-+e%5E%28-lambda%2AT%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lambda+=+0.25 ALIGN=MIDDLE  ALT=\"lambda+=+0.25\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so our function becomes:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28time+%3C=+T%29+=+1+-+e%5E%28-0.25%2AT%29 ALIGN=MIDDLE  ALT=\"P%28time+%3C=+T%29+=+1+-+e%5E%28-0.25%2AT%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now let's answer your questions:\r<BR>\n" );
document.write( "\n" );
document.write( "a) The probability that a call takes 3 minutes or less is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28time+%3C=+3%29+=+1+-+e%5E%28-0.25+%2A+3%29+=+1+-+e%5E%28-0.75%29+=+0.528 ALIGN=MIDDLE  ALT=\"P%28time+%3C=+3%29+=+1+-+e%5E%28-0.25+%2A+3%29+=+1+-+e%5E%28-0.75%29+=+0.528\"   DISABLED_style= \"max-width:90%; height:auto;\" > (to the nearest thousandth)\r<BR>\n" );
document.write( "\n" );
document.write( "This tells us that it is more likely than not that a call will last three minutes or less, even though a typical call is four minutes long.  This result is not as counterintuitive as it looks: longer calls are somewhat rare, but when they happen, their lengths push the typical call length higher than the median call length (the median call length is the call length for which half of all calls are shorter, and half are longer).\r<BR>\n" );
document.write( "\n" );
document.write( "b) The probability that a call takes 4 minutes or less is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28time+%3C=+4%29+=+1+-+e%5E%28-0.25+%2A+4%29+=+1+-+e%5E%28-1%29+=+0.632 ALIGN=MIDDLE  ALT=\"P%28time+%3C=+4%29+=+1+-+e%5E%28-0.25+%2A+4%29+=+1+-+e%5E%28-1%29+=+0.632\"   DISABLED_style= \"max-width:90%; height:auto;\" > (to the nearest thousandth)\r<BR>\n" );
document.write( "\n" );
document.write( "c) The probability that a call takes 5 minutes or less is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28time+%3C=+5%29+=+1+-+e%5E%28-0.25+%2A+5%29+=+1+-+e%5E%28-1.25%29+=+0.713 ALIGN=MIDDLE  ALT=\"P%28time+%3C=+5%29+=+1+-+e%5E%28-0.25+%2A+5%29+=+1+-+e%5E%28-1.25%29+=+0.713\"   DISABLED_style= \"max-width:90%; height:auto;\" > (to the nearest thousandth)\r<BR>\n" );
document.write( "\n" );
document.write( "d) A call length is either less than or equal to 5 minutes, or greater than 5 minutes.  So the sum of the probability that a call takes 5 minutes or less and the probability that a call takes more than 5 minutes should be 1:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%28time+%3C=+5%29+%2B+P%28time+%3E+5%29+=+1+ ALIGN=MIDDLE  ALT=\"+P%28time+%3C=+5%29+%2B+P%28time+%3E+5%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28time+%3E+5%29+=+1+-+P%28time+%3C=+5%29+ ALIGN=MIDDLE  ALT=\"P%28time+%3E+5%29+=+1+-+P%28time+%3C=+5%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28time+%3C=+5%29 ALIGN=MIDDLE  ALT=\"P%28time+%3C=+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%28time+%3E+5%29+=+1+-+%281+-+e%5E%28-1.25%29%29+ ALIGN=MIDDLE  ALT=\"+P%28time+%3E+5%29+=+1+-+%281+-+e%5E%28-1.25%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (substituting using our answer from part (c))<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%28time+%3E+5%29+=+1+-+1+%2B+e%5E%28-1.25%29+ ALIGN=MIDDLE  ALT=\"+P%28time+%3E+5%29+=+1+-+1+%2B+e%5E%28-1.25%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (distributing the negative)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%28time+%3E+5%29+=+e%5E%28-1.25%29+ ALIGN=MIDDLE  ALT=\"+P%28time+%3E+5%29+=+e%5E%28-1.25%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (simplifying)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%28time+%3E+5%29+=+0.287+ ALIGN=MIDDLE  ALT=\"+P%28time+%3E+5%29+=+0.287+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (to the nearest thousandth)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );